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[Solved] In Figure, ar(DRC) = ar(DPC) and ar(BDP) = ar(ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums.

  

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In Figure, ar(DRC) = ar(DPC) and ar(BDP) = ar(ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums.

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Given,

ar(△DRC) = ar(△DPC)

ar(△BDP) = ar(△ARC)

To Prove,

ABCD and DCPR are trapeziums.

Proof:

ar(△BDP) = ar(△ARC)

⇒ ar(△BDP) – ar(△DPC) = ar(△DRC)

⇒ ar(△BDC) = ar(△ADC)

ar(△BDC) = ar(△ADC).

∴ ar(△BDC) and ar(△ADC) are lying in-between the same parallel lines.

∴ AB ∥ CD

ABCD is a trapezium.

Similarly,

ar(△DRC) = ar(△DPC).

∴, ar(△DRC) andar(△DPC) are lying in-between the same parallel lines.

∴ DC ∥ PR

∴ DCPR is a trapezium.

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